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Research Papers

Analysis and Interpretation of Longitudinal Waves in Periodic Multiphase Rods Using the Method of Reverberation-Ray Matrix Combined With the Floquet-Bloch Theorem

[+] Author and Article Information
Y. Q. Guo

State Key Laboratory of Turbulence and
Complex Systems;
College of Engineering,
Peking University,
Beijing 100871, China;
Key Laboratory of Mechanics on Disaster and
Environment in Western China,
Ministry of Education,
School of Civil Engineering and Mechanics,
Lanzhou University,
Lanzhou 730000, China
e-mail: guoyq@lzu.edu.cn

D. N. Fang

State Key Laboratory of Turbulence and
Complex Systems,
College of Engineering,
Peking University,
Beijing 100871, China

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 1, 2012; final manuscript received September 4, 2013; published online October 23, 2013. Assoc. Editor: Michael Leamy.

J. Vib. Acoust 136(1), 011006 (Oct 23, 2013) (13 pages) Paper No: VIB-12-1336; doi: 10.1115/1.4025438 History: Received December 01, 2012; Revised September 04, 2013

The method of reverberation-ray matrix (MRRM) combined with the Floquet–Bloch theorem, which serves as an alternative method, is presented for accurately analyzing longitudinal waves in general periodic multiphase rods. Closed-form dispersion relation of periodic quaternary rods is derived. Based on this relation, the functions of constituent-rod parameters in the formation of longitudinal-wave band structures are analytically revealed. Numerical examples validate the proposed method and indicate the characteristics/applications of all kinds of dispersion curves that include the frequency-wave number spectra, the frequency-wavelength spectra, the frequency-phase velocity spectra, the wave number-phase velocity spectra and the wavelength-phase velocity spectra. The effect of unit-cell layout on the frequency band properties and the functions of constituent-rod parameters in the band structure formation are also illustrated numerically. The analysis and interpretation of longitudinal waves in periodic multiphase rods given in this paper will push forward the design of periodic structures for longitudinal wave filtering/guiding and vibration isolation/control applications.

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Figures

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Fig. 1

The general infinite periodic multiphase rod and the selected unit cell

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Fig. 2

The displacements/forces at the ends of the current and the adjacent unit cells

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Fig. 3

Typical parts of the unit cell: the coordinates and the physical variables

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Fig. 4

The obtained first several wave number-frequency spectra of characteristic longitudinal waves in the periodic binary-I and binary-II rods and their comparisons with the existing counterparts

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Fig. 5

The frequency-wave number spectra of longitudinal waves in the exemplified periodic hexadecimal rod

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Fig. 6

The frequency-wavelength spectra of longitudinal waves in the exemplified periodic hexadecimal rod

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Fig. 7

The frequency-phase velocity spectra of longitudinal waves in the exemplified periodic hexadecimal rod

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Fig. 10

Effect of the cross-sectional area of the constituent rod on the band structure formation

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Fig. 11

Effect of the length of the constituent rod on the band structure formation

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Fig. 12

Effect of the mass density of the constituent rod on the band structure formation

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